Very ample divisors over non-algebraically complete field

Question

For a projective scheme $V$ over an algebraically-closed field $k$ it is a well-known fact fact that a base-point-free linear system of divisors is very ample iff it separates $k$-points and tangent vectors. I have not seen this proven for non-algebraically closed fields, and suspect that it actually false in this case. Is my suspicion correct? If so, what is a counter example?

Answer 1

As Hoot's comment intimates, it's clearly not enough to check on $k$-points and their tangent vectors; you have to check at all points defined over finite extensions of $k$ (or, what amounts to the same thing, all $\overline{k}$-points). If you want to work more scheme-theoretically, this means you have to check at all closed points (and their tangent vectors).